Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds |
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Authors: | Souplet, Philippe Zhang, Qi S. |
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Affiliation: | Laboratoire Analyse, Géométrie et Applications, Institut Galilée, Université Paris-Nord 93430 Villetaneuse, France e-mail: souplet{at}math.univ-paris13.fr Department of Mathematics, University of California Riverside, CA 92521, USA e-mail: qizhang{at}math.ucr.edu |
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Abstract: | We derive a sharp, localized version of elliptic type gradientestimates for positive solutions (bounded or not) to the heatequation. These estimates are related to the ChengYauestimate for the Laplace equation and Hamilton's estimate forbounded solutions to the heat equation on compact manifolds.As applications, we generalize Yau's celebrated Liouville theoremfor positive harmonic functions to positive ancient (includingeternal) solutions of the heat equation, under certain growthconditions. Surprisingly this Liouville theorem for the heatequation does not hold even in Rn without such a condition.We also prove a sharpened long-time gradient estimate for thelog of the heat kernel on noncompact manifolds. 2000 MathematicsSubject Classification 35K05, 58J35. |
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